Complex Analysis, Functional Analysis and Approximation Theory, Proceedings of the Conference on Complex Analysis and Approximation Theory Universidade Estadual de Campinas

307 Pages·1986·5.85 MB·English

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS
AND APPROXIMATION THEORY
NORTH-HOLLAND MATHEMATICS STUDIES 125
Notas de Matematica (11 0)
Editor: Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas,
Rio de Janeiro
and University of Rochester
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD
COMPLEX ANALYSIS,
FUNCTIONAL ANALYSIS
AND APPROXIMATION THEORY
Proceedings of the Conference on
Complex Analysis and Approximation Theory
Universidade Estadual de Campinas, Brazil, 23-27 July, 1984
Edited by
Jorge MUJICA
Universidade Estadual de Campinas
Campinas, Brazil
1986
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD
Elsevier Science Publishers B.V., 1986
@
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,
ortransmitted, in any form orbyanymeans, electronic, mechanical, photocopying, recording or
otherwise, without the priorpermission of the copyright owner.
ISBN: 0 444 87997 8
Publishedby:
ELSEVIER SCIENCE PUBLISHERS B.V.
P.O. Box 1991
1000 BZ Amsterdam
The Netherlands
Sole distributors for the U.S.A. and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
52 Vanderbilt Avenue
NewYork, N.Y. 10017
U.S.A.
Library of Congress Catalogingin-PublicationD ata
Conference on Complex Analysis and Approximation Theory
(1984 : Universidada Estadual de Campinas, Brazil)
Complex analysis, functional analysis, and approxi-
mation theory.
(North-Holland mathematics studies ; 125) (Notas de
matemirtica ; 110)
Includes index.
1. Mathematical analysis--Congresses. 2. Approxima-
tion theory--Congresses. 3. Functional analysis--
.
Congresses. I. Mujica. Jorge, 1946- 11. Title.
111. Series. IV. Series: Notas de matdtica (Rio de
Janeiro, Brazil) ; no. ll0.
QAl.N% no. 110 510 8 C5151 86-4504
cw99.61
ISBN 0-444-8'7997-8
PRINTED IN THE NETHERLANDS
V
FOREWORD
These are the Proceedings of the Conference on Complex Analysis
and ApproximationTheory held at the UniversidadeEstadual de Campinas,
Brazil, from July 23 through July 27, 1984. It contains papers of
research orexpository nature, and is addressed to research workers
and advanced graduate students in mathematics. Some of the papers
are the written and expanded texts of lectures delivered at the con-
ference, whereas others have been included by invitation.
The organizing committee of the conference was formed by Msrio
C. Matos, Joao B. Prolla and Jorqe Mujica (chairman). We gratefully
acknowledge financial support fromthe Conselho Nacional de Desenvol-
vimento Cientifico e Tecnol6gico (CNPq), the Fundayso de Amparo 5
Pesquisa do Estado de Sao Paulo (FAPESP)a nd the UniversidadeEstadual
de Campinas.
We also thank Miss Elda Mortari for her excellent typing of the
manuscript.
Jorge Mujica
Campinas, November 1985
This Page Intentionally Left Blank
vi i
CONTENTS
. . . . . .
V. AURICH, Local analytic geometry in Banach spaces. 1
F. BEATROUS and J. BURBEA, Reproducing kernels and
. . . . . . . .
interpolation of holomorphic functions. 25
J. BLATTER, Metric projections of c onto closed vector
. . . . . . . . . . . . . . . . . . . . . .
sublattices 47
. . . . .
J. F. COLOMBEAU, A new theory of generalized functions 57
. . . . . . .
S. DINEEN, The second dual of a JB* triple system 67
R. DWILEWICZ, Holomorphic approximation in the theory of
. . . . . . . . . . . . . . .
Cauchy-Riemann functions. 71
C. FRANCHETTI, Approximation with subspaces of finite
. . . . . . . . . . . . . . . . . . . . . .
codimension 83
. . . . . . . .
H. G. GARNIR, Microhyperbolic analytic functions 95
A. F. IZE, On a topological method for the analysis of the
asymptotic behavior of dynamical systems and
. . . . . . . . . . . . . . . . . . . . . . .
processes 109
M. C. MATOS, On convolution operators in spaces of entire
. . . . . . . . . .
functions of a given type and order 129
R. MENNICKEN and M. MOLLER, Normal solvability in duals of
. . . . . . . . . . . . . . . . . . . . . . .
LF-spaces 173
L. A. MORAES, A Hahn-Banach extension theorem for some
. . . . . . . . . . . . . . . . .
holomorphic functions 205
L. NACHBIN, A glance at holomorphic factorization and
. . . . . . . . . . . . . . . . . .
uniform holomorphy. 221
viii CONTENTS
P. P. NARAYANASWAMI, Classification of (LF)-spaces by some
. . . . . . . . . . . .
Baire-like covering properties. 247
S. NAVARRO and J. SEGUEL, Nonarchimedean gDF-spaces and
. . . . . . . . . . . . . . . . .
continuous functions. 261
0. W. PAQUES and M. C. ZAINE, Pseudo-convexity, u-convexity
. . . . . . . . . . . . . .
and domains of u-holomorphy 273
D. PISANELLI, The proof of the inversion mapping theorem in
. . . . . . . . . . . . . . . . . . . .
a Banach scale. 281
. . . .
M. VALDIVIA, On certain metrizable locally convex spaces 287
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
INDEX.. 295
COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS
AND APPROXIMATION THEORY, J. Mujica (Editor)
@ Elsevier Science Publishers B.V. (North-Holland), 1986 1
LOCAL ANALYTIC GEOMETRY IN BANACH SPACES
Volker Aurich
Mathematisches Institut
der Ludwig - Maximilians -Universitat
Theresienstrasse 39
D-8000 Miinchen, West Germany
ABSTRACT
After motivating analytic geometry in infinite dimensional spaces
we give a survey on the local theory of SF-analytic sets and holo-
morphic semi-Fredholm maps. Moreover the notion of minimal embedding
codimension is introduced. It allows to derive quantitative results
although the dimension may be infinite.
1. MOTIVATION
Analytic geometry in tn deals with the geometrical properties
of the sets of solutions of analytic equations defined in an open
subset of tn. Since many interesting equationsinanalysis like dif-
ferential and integral equations are defined in infinite dimensional
spaces and are analytic or even polynomial it seems reasonable to
develop a concept of analytic geometry in infinite dimensional topo-
logical vector spaces. Although in applications mostly real solutions
are considered one hopes that as in finite dimensionsthe complex
analytic case yields a simpler and more complete theory. But without
further restrictions this is not true: every compact metric space
occurs as the set of solutions of a complex quadratic polynanialecpa-
tion in a suitable Banach space [ 2 1 . Hence in order to obtain sets
of solutionswithnice geometric properties additional conditions have
to be imposed on the regularity of the equation.
We compile some sufficient conditions in the case of Banach
spaces.
Let E and F be complex Banach spaces and R a domain in E. We
call a map f : R + F hoZornorphic if it is complex Frechet differen-
tiable or equivalently if it is complex analytic. Df(xl denotes the
differential at x. We say that a linear operator T : E F is
+
s;>Zittirig if its kernel and its image are complemented subspaces of